Rational Expressions Calculator
Enter a rational expression (an algebraic fraction) to simplify it by cancelling common factors.
Use / for the fraction bar and ^ for powers, e.g. (x^2-1)/(x-1).
A rational expression is a fraction with polynomials on top and bottom. To simplify it, factor the numerator and denominator, then cancel any common factors. For example, (x² − 1)/(x − 1) factors to (x − 1)(x + 1)/(x − 1) and cancels to x + 1, valid for all x except where the original denominator was zero.
How to Use This Calculator
Enter the algebraic fraction with a / for the fraction bar and ^ for powers. The calculator factors the numerator and denominator, cancels anything they share, and returns the simplified expression.
Rational expressions come up throughout algebra and calculus, and simplifying them first usually makes everything that follows — adding, solving, differentiating — much easier.
How the Calculation Works
The key to simplifying an algebraic fraction is factoring. Once the top and bottom are written as products, any factor appearing in both cancels, exactly as with numerical fractions where 6/9 cancels a common 3 to give 2/3.
The cancelled expression is equal to the original everywhere except at the values that made the original denominator zero. Those values remain excluded from the domain even though they no longer appear in the simplified form.
factor numerator and denominator, then cancel common factors
Worked Example
Simplify (x² − 1)/(x − 1). The numerator is a difference of squares: x² − 1 = (x − 1)(x + 1). So the fraction is (x − 1)(x + 1)/(x − 1).
Cancel the common (x − 1) to get x + 1. This holds for every x except x = 1, where the original expression was undefined.
Assumptions and Limitations
- Simplifies by factoring and cancelling common factors
- Values that made the original denominator zero stay excluded from the domain
- Handles polynomial numerators and denominators
- The result is equivalent to the original except at excluded points
Who Uses a Rational Expressions Calculator
Algebra II and precalculus students use this to simplify a rational expression by cancelling common factors. It is useful for homework and for preparing expressions before adding, multiplying, or solving them. Enter the expression to get the reduced form.
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Open calculator →Frequently Asked Questions
What is a rational expression?
It is a fraction whose numerator and denominator are polynomials, such as (x² − 1)/(x − 1). It is the algebraic version of a numerical fraction.
How do you simplify a rational expression?
Factor the numerator and denominator, then cancel any factor they share. (x² − 1)/(x − 1) factors and cancels to x + 1.
Why do excluded values matter?
Cancelling can hide a value that made the original denominator zero. That value is still not allowed, so the simplified expression equals the original everywhere except there.
Can I add rational expressions with this?
This tool simplifies a single expression. To add them, first find a common denominator — the common denominator calculator helps — then combine and simplify the result.
What if nothing cancels?
Then the expression is already in simplest form. Not every rational expression reduces; it only simplifies when the numerator and denominator share a common factor.
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